26. There are two purses, one containing three sovereigns and a shilling, and the other containing three shillings and a sovereign. A coin is taken from one (it is not known which) and dropped into the other; and then on drawing a coin from each purse, they are found to be two shillings. What are the odds against this happening again if two more are drawn, one from each purse? 27. If a triangle is formed by joining three points taken at random in the circumference of a circle, prove that the odds are 3 to 1 against its being acute-angled. 28. Three points are taken at random on the circumference of a circle: what is the chance that the sum of any two of the arcs so determined is greater than the third? 29. A line is divided at random into three parts, what is the chance that they form the sides of a possible triangle? 30. Of two purses one originally contained 25 sovereigns, and the other 10 sovereigns and 15 shillings. One purse is taken by chance and 4 coins drawn out, which prove to be all sovereigns: what is the chance that this purse contains only sovereigns, and what is the probable value of the next draw from it? 31. On a straight line of length a two points are taken at random; find the chance that the distance between them is greater than b. 32. A straight line of length a is divided into three parts by two points taken at random; find the chance that no part is greater than b. 33. If on a straight line of length a+b two lengths a, b are measured at random, the chance that the common part of these lengths c2 shall not exceed c is where c is less than a or b; also the chance ab, that the smaller length b lies entirely within the larger a is a-b α 34. If on a straight line of length a+b+c two lengths a, b are measured at random, the chance of their having a common part which shall not exceed d is where d is less than either a or b. (c+d) 35. Four passengers, A, B, C, D, entire strangers to each other, are travelling in a railway train which contains first-class, m second-class, and n third-class compartments. A and B are gentlemen whose respective a priori chances of travelling first, second, or third class are represented in each instance by λ, u, v; C and D are ladies whose similar a priori chances are each represented by l, m, n. Prove that, for all values of λ, μ, v (except in the particular case when λ : μ: v=l: m : n), A and B are more likely to be found both in the company of the same lady than each with a different one, CHAPTER XXXIII. DETERMINANTS. 485. THE present chapter is devoted to a brief discussion of determinants and their more elementary properties. The slight introductory sketch here given will enable a student to avail himself of the advantages of determinant notation in Analytical Geometry, and in some other parts of Higher Mathematics; fuller information on this branch of Analysis may be obtained from Dr Salmon's Lessons Introductory to the Modern Higher Algebra, and Muir's Theory of Determinants. 486. Consider the two homogeneous linear equations a1x+b1y = 0, multiplying the first equation by b,, the second by b,, subtracting and dividing by x, we obtain and the expression on the left is called a determinant. It consists of two rows and two columns, and in its expanded form each term is the product of two quantities; it is therefore said to be of the second order. The letters a1, b1, a, b, are called the constituents of the determinant, and the terms a,b,, a,b, are called the elements. it follows that the value of the determinant is not altered by changing the rows into columns, and the columns into rows. that is, if we interchange two rows or two columns of the determinant, we obtain a determinant which differs from it only in sign. 489. Let us now consider the homogeneous linear equations a1x + b1y + c1z = 0, By eliminating x, y, z, we obtain as in Ex. 2, Art. 16, a ̧ (b ̧¤ ̧− b ̧o ̧) + b ̧ (e̟ ̧à ̧− e̟ ̧a ̧)+c ̧(ab ̧—à ̧b ̧) = 0, 1 3 2 3 and the expression on the left being a determinant which consists of three rows and three columns is called a determinant of the third order. 490. By a rearrangement of terms the expanded form of the above determinant may be written that is, the value of the determinant is not altered by changing the rows into columns, and the columns into rows. We shall now explain a simple method of writing down the expansion of a determinant of the third order, and it should be noticed that it is immaterial whether we develop it from the first row or the first column. From equation (1) we see that the coefficient of any one of the constituents a1, a,, a, is that determinant of the second order which is obtained by omitting the row and column in which it occurs. These determinants are called the Minors of the original determinant, and the left-hand side of equation (1) may be written where A,, 4, 4, are the minors of a,, a, a, respectively. Again, from equation (2), the determinant is equal to where A, B, C, are the minors of a,, b, c, respectively. Thus it appears that if two adjacent columns, or rows, of the determinant are interchanged, the sign of the determinant is changed, but its value remains unaltered. If for the sake of brevity we denote the determinant by (a,b,c), then the result we have just obtained may be written (b ̧α ̧c ̧) = − (a ̧b ̧ç ̧). Similarly we may shew that (c ̧α ̧b ̧) = − (a‚ç‚b ̧) =+ (a,b,c ̧). 493. If two rows or two columns of the determinant are identical the determinant vanishes. For let D be the value of the determinant, then by interchanging two rows or two columns we obtain a determinant whose value is -D; but the determinant is unaltered; hence D= – D, that is D=0. Thus we have the following equations, 494. If each constituent in any row, or in any column, is multiplied by the same factor, then the determinant is multiplied by that factor. |